Sampling Conditions for Clipping-free Voronoi Meshing by the VoroCrust Algorithm

Decomposing the volume bounded by a closed surface using simple cells is a key step in many applications. Although tetrahedral meshing is well-established with several successful implementations, many research questions remain open for polyhedral meshing, which promises significant advantages in some contexts. Unfortunately, current approaches to polyhedral meshing often resort to clipping cells near the boundary, which results in certain defects. In this talk, we present an analysis of the VoroCrust algorithm, which leverages ideas from α-shapes and the power crust algorithm to produce conforming Voronoi cells on both sides of the surface. We derive sufficient conditions for a weighted sampling to produce a topologically-correct and geometrically-accurate reconstruction, using the -sampling paradigm with a standard sparsity condition. The resulting surface reconstruction consists of weighted Delaunay triangles, except inside tetrahedra with a negative weighted circumradius where a Steiner vertex is generated close to the surface. Generating quality meshes is an important problem in computer graphics and geometric modeling. There has been a growing interest in polyhedral meshing as it promises significant advantages over tetrahedral or hex-dominant meshing in some contexts. For exmaple, polyhedral meshing offers higher degrees of freedom per element and is more efficient in filling a space, because it produces fewer elements for the same number of vertices. Within the class of polyhedral mesh elements, Voronoi cells enjoy several geometric properties, e.g., planar facets and positive Jacobians, which make them particularly suitable for numerical simulations. A conforming mesh exhibits two desirable properties simultaneously : 1) a decomposition of the enclosed volume, and 2) a reconstruction of the bounding surface. A common technique for producing boundary-conforming decomposition from Voronoi cells relies on clipping, i.e., intersecting and truncating, each cell by the bounding surface [3]. An alternative to clipping is to locally mirror the Voronoi generators on either side of the surface [2]. VoroCrust can be viewed as a principled mirroring technique. Similar to the power crust [1], the reconstruction is composed of the facets shared by cells on the inside and outside of the manifold. However, VoroCrust uses pairs of unweighted generators tightly hugging the surface, which allows further decomposition of the interior without disrupting the surface reconstruction. VoroCrust can also be viewed as a special case of the weighted α-shape [4]. A description of the abstract VoroCrust algorithm we analyze is provided next. Figure 1 illustrates the basic concepts in 2D.